Xiang Zhu, a client of FinCorp Inc., has phoned you with a question. She has been reading a finance textbook and cannot understand how to use the binomial option pricing model to value a call option. The underlying stock is currently trading for $100. There is a 30-percent chance it will increase to $190 in one year, and a 70-percent chance it will fall to $85 in the same period. The risk-free rate is 10 percent. There is a call option with a strike price of $170, which according to the binomial model should have a value of $4.329. Xiang is confused because she calculates that if there is a 30-percent chance the call will be worth $20 and a 70-percent chance it will be worth zero, the expected value should be $6. Why is it only worth $4.329?
a. Demonstrate that if the call was trading for $6, there would be an arbitrage opportunity.
b. Calculate the risk-neutral probabilities.
c. Calculate the expected present value of the call option using the risk-neutral probabilities.
d. Why can we value options as if the investors are risk neutral?